Function by properties

Question

I am looking for a function $f(x)$, which

1. is vertical at the origin ($f'(0)\approx -\infty$)
2. goes to zero further from the origin ($f(x) \to 0$ for $x\to \infty$ and for $x\to -\infty$)
3. and is rotationally symmetric ($f(x) = -f(-x)$ for all $x$)
4. is finite ($\forall x: -\infty \neq f(x)\neq \infty$)
5. is differentiable ($f'(x)$ is not undefined for every $x$)

(visually this will look like one heart beat on a monitor, just to give some idea)

Of course I'd like the function to be the least complex as possible. (Furthermore it would be nice if the function would have some statistical importance.)

Do you know any function satisfying these properties?

How does one go about searching for a function, which has to satisfy such properties?

Is there any application which helps you in finding the right/best function? (Mathematica?)

Answer 1

How about $-x^\frac{1}{3} \, e^{-x^2}$?

Answer 2

How about $$ f(x)=-\text{sgn}(x)\,\frac{\sqrt{|x|}}{1+x^2}\text{ ?} $$ Here $\text{sgn}(x)=1$ if $x>0$, $-1$ if $x<0$ and $0$ if $x=0$. It is not differentiable at $x=0$, but you wanted $f'(0)\approx-\infty$.

More general example:

$$ f(x)=-\text{sgn}(x)\,\frac{|x|^p}{1+|x|^q}\quad 0<p<1,\quad q>p. $$